How to combine/manipulate two summations into one summation in general?

Question

I always struggle to understand what I can and can't do with sums. In fact, even when convergence isn't an issue, I still get confused. What can I do about this problem?

For instance, I am currently trying to show that the multiplication of (formal) polynomials is associative. By polynomial over a field $F$, let us mean a function $a : \mathbb{N} \rightarrow F$, where $F$ is a field, such that for $n$ sufficiently large it holds that $a_n = 0$. Multiplication of polynomials can be defined as follows. For all polynomials $a$ and $b$ and all $n \in \mathbb{N}$, it holds that $$(ab)_n = \sum_{i,j \in \mathbb{N}}^{i+j=n} a_i b_j.$$

So the problem is to show that for all polynomials $a$, $b$ and $c$ it holds that $(ab)c=a(bc)$. Fix any polynomials $a$, $b$ and $c$

Then $$[(ab)c]_n = \sum_{m,k \in \mathbb{N}}^{m+k=n} (ab)_m c_k = \sum_{m,k \in \mathbb{N}}^{m+k=n} (\sum_{i,j \in \mathbb{N}}^{i+j=m} a_i b_j) c_k = \sum_{m,k \in \mathbb{N}}^{m+k=n} \sum_{i,j \in \mathbb{N}}^{i+j=m} a_i b_j c_k.$$

Now I want to combine the two sums into a single sum $\displaystyle \sum_{i,j,k \in \mathbb{N}}^{i+j+k=n}$. How do I justify this?

In general, how does one justify these kinds of things?

Answer 1

Here’s how you can do it using only the most common form of summation notation.

$$\begin{align*} [(ab)c]_n&=\sum_{m,k\in\Bbb N}^{m+k=n}(ab)_mc_k\\\\ &=\sum_{m=0}^n(ab)_mc_{n-m}\\\\ &=\sum_{m=0}^n\left(\sum_{i=0}^ma_ib_{m-i}\right)c_{n-m}\\\\ &=\sum_{m=0}^n\sum_{i=0}^ma_ib_{m-i}c_{n-m}\\\\ &=\sum_{i=0}^n\sum_{m=i}^na_ib_{m-i}c_{n-m}\\\\ &=\sum_{i=0}^na_i\sum_{m=i}^nb_{m-i}c_{n-m}\\\\ &=\sum_{i=0}^na_i\sum_{k=0}^{n-i}b_kc_{n-i-k}&&(k=m-i)\\ &=\sum_{i=0}^na_i(bc)_{n-i}\\\\ &=[a(bc)]_n \end{align*}$$